Histograms, How to Draw a Histogram

Histograms

A histogram is a representation of data using bars of equal width and different heights.

Histogram is represented for grouped frequency distribution. In a histogram, we draw bars of uniform width adjacent to each other. All the class intervals in a histogram are taken along X-axis and their respective frequencies on Y-axis.

Let us draw the histogram of the following data.

Marks	Number of students
40 – 50	1
50 – 60	2
60 – 70	5
70 – 80	12
80 – 90	10
90 – 100	20
100 – 110	22
110 – 120	14
120 – 130	6
130 – 140	4

It is similar to a bar graph but a histogram groups numbers into ranges.

The height of each bar shows how many fall into each range.

And you decide what ranges to use!

What is a Histogram?

A histogram is a plot that lets you discover and show the underlying frequency distribution of a set of continuous data. An example of a histogram using the raw data is constructed below:

36	25	38	46	55	68	72	55	36	38
67	45	22	48	91	46	52	61	58	55

To construct a histogram from a continuous variable you first need to split the data into intervals, called bins. In the example above, marks has been split into bins, with each bin representing a 10-mark period starting at 20 marks. Each bin contains the number of occurrences of marks in the data set that are contained within that bin. For the above data set, the frequencies in each bin have been tabulated along with the marks that contributed to the frequency in each bin:

Marks	Frequency	Marks Included in Bin
20-30	2	25,22
30-40	4	36,38,36,38
40-50	4	46,45,48,46
50-60	5	55,55,52,58,55
60-70	3	68,67,61
70-80	1	72
80-90	0	-
90-100	1	91

Notice that, unlike a bar chart, there are no "gaps" between the bars (although some bars might be "absent" reflecting no frequencies e.g. for 80-90 in the above example). This is because a histogram represents a continuous data set, and as such, there are no gaps in the data.

When to Use a Histogram?

Use a histogram when:

· The data are numerical

· You want to see the shape of the data’s distribution, especially when determining whether the output of a process is distributed approximately normally

· Analyzing whether a process can meet the customer’s requirements

· Analyzing what the output from a supplier’s process looks like

· Seeing whether a process change has occurred from one time period to another

· Determining whether the outputs of two or more processes are different

· You wish to communicate the distribution of data quickly and easily to others

Difference Between a Histogram and a Bar Graph

1. The histogram is drawn for grouped frequency distribution while a bar graph is drawn for ungrouped frequency distribution.

2. There is no gap between two bars of a histogram while there is a uniform gap between two bars of a bar graph.

How to Draw a Histogram?

Solved Examples on Histogram

Example 1: Draw a histogram for the given data.

Class Interval	Frequency
20 – 30	8
30 – 40	5
40 – 50	6
50 – 60	3
60 – 70	4
70 – 80	5
80 – 90	4
90 – 100	3

Solution: Draw two axes perpendicular to each other and then draw the rectangles of equal width and of heights equal to the given frequencies.

Example 2: The given histogram depicts the marks obtained by 40 students of a class in an examination.

Now, answer the following questions.

a. What is the size of each class interval?

b. How many students scored marks less than 20?

c. If the passing mark is 40, how many students failed?

d. What is the interval of highest marks and how many students obtained marks in this interval?

Solution:

a. The class intervals are 0–20, 20–40, etc. So the class size is 20 (20–0 = 20 or 40 – 20 = 20).

b. 2 students scored marks less than 20.

c. 6 students failed (2 students from the interval 0–20 and 4 students from the interval 20– 40).

d. The interval of highest marks is 80–100 and 3 students obtained marks in this interval.

Example 3: The following table shows the heights of 50 students.

Height (in cm)	Frequency
150 – 155	7
155 – 160	12
160 – 165	18
165 – 170	10
170 – 175	3

a. Represent the data using a histogram.

b. Describe the shape of the histogram.

Solution: a. The histogram is shown as follows.

b. The histogram shows that the heights of most students lie between 160 cm and 165 cm. There are more students in the lowest class than in the highest class.

Example 4: The given histogram shows the number of hours 30 students spend outdoor on a holiday.

a. What is the size of each class interval?

b. How many students spend less than an hour outdoors?

c. In which time interval did the largest number of students spend time outdoors?

d. How many students spend four or more hours outdoors?

Solution: a. The class intervals are 1 – 2, 2 – 3, 3 – 4, …

Thus, class size = 2 – 1 = 1 hour

b. 2 students spend less than an hour outdoors on a holiday.

c. The largest number of students spend 1 – 2 hours outdoors.

d. 2 students spend four or more hours outdoors.

Related Topics:

How to draw a pictograph?

How to draw a bar graph?

How to draw a pie chart?

How to draw a line graph?